1 September 2004 Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model A. Introduction and assumptions The classical normal linear regression model can be written as (1) or (2) where xtN is the tth row of the matr
Chapter 4
Prediction, Goodness-of-fit, and
Modeling Issues
Walter R. Paczkowski
Rutgers University
Principles of Econometrics, 4th
Edition
Chapter 4: Prediction, Goodness-of-fit, and Modeling Issues
Page 1
Chapter Contents
4.1 Least Square Prediction
4.
Chapter 10
Random Regressors and
Moment-Based Estimation
Principles of Econometrics, 4th
Edition
Chapter 10: Random Regressors and
Moment-Based Estimation
Page 1
Chapter Contents
10.1 Linear Regression with Random xs
10.2 Cases in Which x and e are Corr
Chapter 9
Regression with Time Series
Data:
Stationary Variables
Walter R. Paczkowski
Rutgers University
Principles of Econometrics, 4th
Edition
Chapter 9: Regression with Time Series Data:
Stationary Variables
Page 1
Chapter Contents
9.1 Introduction
9
Regression Motivation
Dan Yavorsky
1/12
Plan
I want to show you something that is mathematically more general
than what is done in class.
But then I will tie the general things specifically to linear
regression.
I do this because:
1. it helped me with que
IV. Introduction to Time Series AR(1) Model and Forecasting
a.
b.
c.
d.
e.
f.
g.
Introduction to Dependent Observations
Checking for Independence
Autocorrelation
The AR(1) Model
Random Walks
Trend Models and US GDP
Google Trend Modelling
Chapter IV. Slide
II. The Multiple Regression Model
a.
b.
c.
d.
e.
f.
g.
The Multiple Regression Model
The Data and Least Squares
Inference and F-tests
Prediction
Multiple Regression Explained: The Pricing Example
More on the Interpretation of MR Coefficients
Multi-factor
V. Advanced Regression Topics
a.
b.
c.
d.
e.
f.
g.
h.
MR Standard Errors
Multi-colinearity
Standardized Residuals, Leverage, and Outliers
Nonlinearity
Dummy Variables
Heteroskedasticity
Bootstrapping Regression Models
Model Selection: The Equity Premium P
VI. Likelihood and Maximum Likelihood Estimation
a.
b.
c.
d.
e.
The Likelihood Function for Regression Models
Method of Maximum Likelihood
Properties of MLEs
ARCH-M Likelihood and Example
Conclusions Regarding Maximum Likelihood
Chapter VI. Slide 1
a. Lik
Math-Stat Review
Dan Yavorsky
1/18
Random Variables
Given an experiment, denote an outcome s, where s S such
that S is the collection of all possible outcomes
Definition:
I
A function X : S R, which assigns to each element s S
one (and only one) number X(
Relaxing OLS Assumptions
Dan Yavorsky
1/13
Regression
In general regression tries to explain Y with X:
Y = f (X) +
Very often, X is assumed to be linear in the parameters :
Y = f (X) +
In Linear Regression, that formula is:
Y = X +
2/13
OLS Assumptions
Key Statistical and Mathematical Prerequisites
NOTE TO THE STUDENT:
This document is designed to review previously acquired statistics knowledge. It is not
designed to teach this material from scratch. If you arent familiar with the basic concepts,
please
Tutorial: ggplot2
Ramon Saccilotto
Universittsspital Basel
Hebelstrasse 10
T 061 265 34 07 F 061 265 31 09 [email protected]
www.ceb-institute.org
Basel Institute for Clinical Epidemiology and Biostatistics
About the ggplot2 Package
Introduction
"ggplot
Chapter 7
Using Indicator Variables
Walter R. Paczkowski
Rutgers University
Principles of Econometrics, 4th
Edition
Chapter 7: Using Indicator Variables
Page 1
Chapter Contents
7.1 Indicator Variables
7.2 Applying Indicator Variables
7.3 Log-linear Mod
Chapter 8
Heteroskedasticity
Walter R. Paczkowski
Rutgers University
Principles of Econometrics, 4th
Edition
Chapter 8: Heteroskedasticity
Page 1
Chapter Contents
8.1 The Nature of Heteroskedasticity
8.2 Detecting Heteroskedasticity
8.3 Heteroskedastic
Economics 671 Problem Set #4 Convergence Concepts
2 (1) Suppose that Xn a for some constant a and consider the sequence Yn = Xn . Show directly
p
[that is, without making use of the theorem discussed in class that g(Xn ) g(a) for a continuous function g]
Economics 671 Solutions: Problem Set #2 Special Distributions and Changes of Variable
(1) MATLAB code for this portion of the problem set has been provided. Note that the inverse transform method suggests that u, where u U (0, 1) will provide a draw from
Economics 573 Problem Set 5 Fall 2002 Due: 4 October 2002 1. In random sampling from any population with E(X) = : and Var(X) = F2, show (using Chebyshev's inequality) that sample mean converges in probability to :. 2. In random sampling from any populatio
Economics 573 Problem Set 4 Fall 2002 Due: 20 September 1. Ten students selected at random have the following "final averages" in physics and economics. Students Physics Economics a. 1 66 75 2 70 70 3 50 65 4 80 88 5 60 60 6 70 85 7 55 60 8 90 97 9 75 82
Economics 671 Problem Set #1 Univariate Probability
(1) Casella and Berger, 2.24: Compute E(x) and Var(X) for each of the following probability distributions (a) f (x) = axa-1 , 0 < x < 1, a > 0.
1 (b) f (x) = n , x = 1, 2, , n where n is an integer.
(Not
Economics 671 Solutions: Problem Set #4 Convergence Concepts
(1) Fix
> 0. We seek to show that
n 2 lim Pr |Xn - a2 | >
= 0.
Choose a > 0 such that 2 + 2|a| <
2 Pr |Xn - a2 | >
and write the above probability as
2 2 = Pr |Xn - a2 | > , |Xn - a| + Pr |Xn -
Economics 671 Solutions: Problem Set #2 Multivariate and Conditional Probability
(1) (a) First, note that the density is always positive over its support. Moreover,
0 x
2 exp[(x + y)]dydx = 1
(which you can verify through some rather simple integration)
Economics 671 Problem Set #3 Special Distributions and Changes of Variable
(1a) Suppose that f (x) = 2x, 0 x 1.
Describe how you can generate draws from this distribution. (1b) Using MATLAB and your solution in (1a), write a program that generates 10,000
Economics 671 Problem Set #5 Central Limit Theorem
(1) Suppose that X1 , X2 , , Xn denote an iid sample from an exponential distribution with density function: f (Xi ) = 1 exp(xi 1 ), xi 0 i. (1a) Using the moment generating function approach, derive the
Economics 671
Problem Set #2
Multivariate and Conditional Probability
(1) Consider the joint density function
f (x, y) =
2 exp([x + y])
0
for 0 x y, 0 y
otherwise .
(a) Show that this is a valid bivariate density function.
(b) Derive the marginal densitie
University of New Mexico
Economics Department
ECON-309 Introduction to Statistics and Econometrics
Assignment 2
Instructions: This assignment is due on Friday, September 9th. You may submit through Learn up until midnight, submit a paper
copy in class or
Introduction to Rossis 237Q Slides
I have made this longer than usual, because I lack the time to make it
short. - Pascal
I had the time!
A lot of work has gone into simplifying and stripping down to the
essence. This means that the slides will require ca